Splitting up a Texture and Object into different parts
Separating Texture Maps

You can bake textures from one object to another. If the objects are already separated (as it seems to be in your case) it makes things easier. 

1. Separating the Object to which you want to bake the new texture.


Tab into Edit Mode and select all faces of the mesh you want to have in a separate object. 
hit Shift+D to duplicate the selection.
hit P and select "Selection" to create a new object from your selected faces.




2. Preparing new Objct


Select your newly created object, (tab into object mode, if otherwise) go to "Properties - Materials" and delete the material.





With you newly created object, tab into Edit Mode and create a new texture of the size you desire (also pack the islands and correct the margins, if you want)*.




3. Bake texture from Original to new Object


Select the original object.
Shift>+Select the new Object.
Go to "Properties - Render - Bake", select "Textures", turn on "Selected to Active" and hit "Bake". 
Don't forget to save the new texture map after baking.




Create a new material for your newly created object and assign the Texture(s) to it. You can then go to the original object and delete the part from which we created the copy in Step 1. 

*Be aware, that changing the UV size as well as the size of the new texture might result in visible texture differences between the original and the new object.